Electronic Transactions on Numerical Analysis
Volume 39, 2012
Contents
1
The MR3 -GK algorithm for the bidiagonal SVD. Paul R. Willems and Bruno Lang.
Abstract.
Determining the singular value decomposition of a bidiagonal matrix is a frequent
subtask in numerical computations. We shed new light on a long-known way to utilize the algorithm of multiple relatively robust representations, MR3 , for this task
by casting the singular value problem in terms of a suitable tridiagonal symmetric
eigenproblem (via the Golub–Kahan matrix). Just running MR3 “as is” on the tridiagonal problem does not work, as has been observed before (e.g., by B. Großer and
B. Lang [Linear Algebra Appl., 358 (2003), pp. 45–70]). In this paper we give more
detailed explanations for the problems with running MR3 as a black box solver on
the Golub–Kahan matrix. We show that, in contrast to standing opinion, MR3 can
be run safely on the Golub–Kahan matrix, with just a minor modification. A proof
including error bounds is given for this claim.
Key Words.
bidiagonal matrix, singular value decomposition, MRRR algorithm, theory and implementation, Golub–Kahan matrix
AMS Subject Classifications.
65F30, 65F15, 65G50, 15A18
22
Collocation methods based on radial basis functions for the coupled Klein-GordonSchrödinger equations. Ahmad Golbabai and Ali Safdari-Vaighani.
Abstract.
This paper presents radial basis function (RBF) collocation methods for the coupled
Klein-Gordon-Schrödinger equations. Unlike traditional mesh oriented methods,
RBF collocation methods require only a scattered set of nodes in the domain where
the solution is approximated. For the RBF collocation method in finite difference
mode (RBF-FD), weights for the finite difference formula are obtained by solving
local RBF interpolation problems set up around each node in the computational domain. We show that the RBF-FD method has good accuracy and a sparse coefficient
matrix, as compared to the global form of the RBF collocation method and other
methods.
Key Words.
collocation methods, coupled Klein-Gordon-Schrödinger equations, radial basis
functions (RBF)
AMS Subject Classifications.
65N06, 65N40, 65D25, 35Q51
i
32
A combined fourth-order compact scheme with an accelerated multigrid method for
the energy equation in spherical polar coordinates. T. V. S. Sekhar, R. Sivakumar,
S. Vimala, and Y. V. S. S. Sanyasiraju.
Abstract.
A higher-order compact scheme is combined with an accelerated multigrid method
to solve the energy equation in a spherical polar coordinate system. The steady
forced convective heat transfer from a sphere which is under the influence of an
external magnetic field is simulated. The convection terms in the energy equation
are handled in a comprehensive way avoiding complications in the calculations. The
angular variation of the Nusselt number and mean Nusselt number are calculated
and compared with recent experimental results. Upon applying the magnetic field, a
slight degradation of the heat transfer is found for moderate values of the interaction
parameter N, and for high values of N an increase in the heat transfer is observed
leading to a nonlinear behavior. The speedy convergence of the solution using the
multigrid method and accelerated multigrid method is illustrated.
Key Words.
higher-order compact scheme, accelerated multigrid method, forced convection heat
transfer, external magnetic field
AMS Subject Classifications.
65N06, 65N55, 35Q80
46
Extremal interpolatory problem of Fejér type for all classical weight functions.
Przemysław Rutka and Ryszard Smarzewski.
Abstract.
Several constructive solutions of interpolating problems of Fejér, Egerváry and
Turán, connected with the optimal, most economical and stable interpolation are
known for Jacobi, Hermite and Laguerre orthogonal polynomials. In this paper we
solve the interpolatory weighted problem of Fejér type for all positive solutions of
the Pearson differential equation, which generate finite or infinite sequences of the
classical orthogonal polynomials. More precisely, we establish that the Fejér problem is generic in this class of polynomials and present an elementary unified proof of
this fact. Next, these results are used to establish a complete solution of the Egerváry
and Turán interpolatory problem.
Key Words.
Fejér extremal problem, Pearson differential equation, Sturm-Liouville differential equation, classical orthogonal polynomials, Lagrange optimal interpolation,
weighted stability, most economical systems
AMS Subject Classifications.
41A05, 42C05, 65D05, 49K35
60
On domain-robust preconditioners for the Stokes equations. Manfred Dobrowolski.
Abstract.
It is well known that the LBB-constant of the Stokes equations and its discrete
counterpart degenerate on domains with high aspect ratio. For the solution of the
corresponding linear system we propose two preconditioners that are proven to be
ii
independent of the aspect ratio of the underlying domain. Both preconditioners are
based on a refined approximation of the Schur complement using a coarser grid.
Key Words.
Stokes equations, LBB constant, finite elements, mixed methods
AMS Subject Classifications.
65N15, 65N30, 35Q30, 76D07
75
The complete stagnation of GMRES for n ≤ 4. Gérard Meurant.
Abstract.
We study the problem of complete stagnation of the generalized minimum residual
method for real matrices of order n ≤ 4 when solving nonsymmetric linear systems
Ax = b. We give necessary and sufficient conditions for the non-existence of a real
right-hand side b such that the iterates are xk = 0, k = 0, . . . , n − 1, and xn = x.
We illustrate these conditions with numerical experiments. We also give a sufficient
condition for the non-existence of complete stagnation for a matrix A of any order
n.
Key Words.
GMRES, stagnation, linear systems
AMS Subject Classifications.
15A06, 65F10
102
Trigonometric Gaussian quadrature on subintervals of the period. Gaspare Da Fies
and Marco Vianello.
Abstract.
We construct a quadrature formula with n + 1 angles and positive weights which is
exact in the (2n + 1)-dimensional space of trigonometric polynomials of degree ≤ n
on intervals with length smaller than 2π. We apply the formula to the construction of
product Gaussian quadrature rules on circular sectors, zones, segments, and lenses.
Key Words.
trigonometric Gaussian quadrature, subintervals of the period, product Gaussian
quadrature, circular sectors, circular zones, circular segments, circular lenses
AMS Subject Classifications.
65D32
113
Boundary element collocation method for solving the exterior Neumann problem
for Helmholtz’s equation in three dimensions. Andreas Kleefeld and Tzu-Chu Lin.
Abstract.
We describe a boundary integral equation that solves the exterior Neumann problem
for the Helmholtz equation in three dimensions. The unique solution is found by
approximating a Fredholm integral equation of the second kind with the boundary
element collocation method. We prove superconvergence at the collocation points,
iii
distinguishing the cases of even and odd interpolation. Numerical examples demonstrate the performance of the method solving the integral equation and confirm the
superconvergence.
Key Words.
Fredholm integral equation of the second kind, Helmholtz’s equation, exterior Neumann problem, boundary element collocation method, superconvergence
AMS Subject Classifications.
35J05, 45B05, 65N35, 65N38
144
Estimations of the trace of powers of positive self-adjoint operators by extrapolation
of the moments. Claude Brezinski, Paraskevi Fika, and Marilena Mitrouli.
Abstract.
Let A be a positive self-adjoint linear operator on a real separable Hilbert space H.
Our aim is to build estimates of the trace of Aq , for q ∈ R. These estimates are
obtained by extrapolation of the moments of A. Applications of the matrix case are
discussed, and numerical results are given.
Key Words.
trace, positive self-adjoint linear operator, symmetric matrix, matrix powers, matrix
moments, extrapolation
AMS Subject Classifications.
65F15, 65F30, 65B05, 65C05, 65J10, 15A18, 15A45
156
Spectral deflation in Krylov solvers: A theory of coordinate space based methods.
Martin H. Gutknecht.
Abstract.
For the iterative solution of large sparse linear systems we develop a theory for a
family of augmented and deflated Krylov space solvers that are coordinate based in
the sense that the given problem is transformed into one that is formulated in terms
of the coordinates with respect to the augmented bases of the Krylov subspaces. Except for the augmentation, the basis is as usual generated by an Arnoldi or Lanczos
process, but now with a deflated, singular matrix. The idea behind deflation is to
explicitly annihilate certain eigenvalues of the system matrix, typically eigenvalues
of small absolute value. The deflation of the matrix is based on an either orthogonal
or oblique projection on a subspace that is complimentary to the deflated approximately invariant subspace. While an orthogonal projection allows us to find minimal
residual norm solutions, the oblique projections, which we favor when the matrix is
non-Hermitian, allow us in the case of an exactly invariant subspace to correctly deflate both the right and the corresponding left (possibly generalized) eigenspaces of
the matrix, so that convergence only depends on the non-deflated eigenspaces. The
minimality of the residual is replaced by the minimality of a quasi-residual. Among
the methods that we treat are primarily deflated versions of GMR ES, M IN R ES, and
QMR, but we also extend our approach to deflated, coordinate space based versions
iv
of other Krylov space methods including variants of CG and B I CG. Numerical results will be published elsewhere.
Key Words.
linear equations, Krylov space method, Krylov subspace method, deflation, augmented basis, recycling Krylov subspaces, (singular) preconditioning, GMR ES,
M IN R ES, QMR, CG, B I CG
186
Spectral analysis of a block-triangular preconditioner for the Bidomain system in
electrocardiology. Luca Gerardo-Giorda and Lucia Mirabella.
Abstract.
In this paper we analyze in detail the spectral properties of the block-triangular
preconditioner introduced by Gerardo-Giorda et al. [J. Comput. Phys., 228 (2009),
pp. 3625-3639] for the Bidomain system in non-symmetric form. We show that the
conditioning of the preconditioned problem is bounded in the Fourier space independently of the frequency variable, ensuring quasi-optimality with respect to the mesh
size. We derive an explicit formula to optimize the preconditioner performance by
identifying a parameter that depends only on the coefficients of the problem and is
easy to compute. We provide numerical tests in three dimensions that confirm the
optimality of the parameter and the substantial independence of the mesh size.
Key Words.
electrocardiology, Bidomain system, preconditioning, finite elements
AMS Subject Classifications.
65M60, 65M12, 65F08
202
Creating domain mappings. Kendall Atkinson and Olaf Hansen.
Abstract.
1−1
Consider being given a mapping ϕ : S d−1 −→ ∂Ω, with ∂Ω the (d − 1)onto
dimensional smooth boundary surface for a bounded open simply-connected region Ω in Rd , d ≥ 2. We consider the problem of constructing an extension
1−1
Φ : B d −→ Ω with Bd the open unit ball in Rd . The mapping is also required
onto
to be continuously differentiable with a non-singular Jacobian matrix at all points.
We discuss ways of obtaining initial guesses for such a mapping Φ and of then improving it by an iteration method.
Key Words.
domain mapping, multivariate polynomial, constrained minimization, nonlinear iteration
AMS Subject Classifications.
65D99
231
A robust FEM-BEM MinRes solver for distributed multiharmonic eddy current
optimal control problems in unbounded domains. Michael Kolmbauer.
Abstract.
This work is devoted to distributed optimal control problems for multiharmonic eddy
current problems in unbounded domains. We apply a multiharmonic approach to the
optimality system and discretize in space by means of a symmetrically coupled finite and boundary element method, taking care of the different physical behavior in
v
conducting and non-conducting subdomains, respectively. We construct and analyze
a new preconditioned MinRes solver for the system of frequency domain equations.
We show that this solver is robust with respect to the space discretization and time
discretization parameters as well as the involved “bad” parameters like the conductivity and the regularization parameters. Furthermore, we analyze the asymptotic
behavior of the error in terms of the discretization parameters for our special discretization scheme.
Key Words.
time-periodic optimization, eddy current problems, finite element discretization,
boundary element discretization, symmetric coupling, MinRes solver
AMS Subject Classifications.
49N20, 35Q61, 65M38, 65M60, 65F08
253
Improved predictor schemes for large systems of linear ODEs. Mouhamad Al Sayed
Ali and Miloud Sadkane.
Abstract.
When solving linear systems of ordinary differential equations (ODEs) with constant
coefficients by implicit schemes such as implicit Euler, Crank-Nicolson, or implicit
Runge-Kutta, one is faced with the difficulty of correctly solving the repeated linear
systems that arise in the implicit scheme. These systems often have the same matrix
but different right-hand sides. When the size of the matrix is large, iterative methods
based on Krylov subspaces can be used. However, the effectiveness of these methods
strongly depends on the initial guesses. The closer the initial guesses are to the exact
solutions, the faster the convergence. This paper presents an approach that computes
good initial guesses to these linear systems. It can be viewed as an improved predictor method. It is based on a Petrov-Galerkin process and multistep schemes and
consists of building, throughout the iterations, an approximation subspace using the
previous computations, where good initial guesses to the next linear systems can be
found. It is shown that the quality of the computed initial guess depends only on the
stepsize of the discretization and the dimension of the approximation subspace. The
approach can be applied to most of the common implicit schemes. It is tested on
several examples.
Key Words.
convergence acceleration, implicit scheme, predictor, Petrov-Galerkin, GMRES
AMS Subject Classifications.
65L20, 65F10
271
Variational ensemble Kalman filtering using limited memory BFGS. Antti Solonen, Heikki Haario, Janne Hakkarainen, Harri Auvinen, Idrissa Amour, and
Tuomo Kauranne.
Abstract.
The extended Kalman filter (EKF) is one of the most used nonlinear state estimation
methods. However, in large-scale problems, the CPU and memory requirements of
EKF are prohibitively large. Recently, Auvinen et al. proposed a promising approximation to EKF called the variational Kalman filter (VKF). The implementation of
VKF requires the tangent linear and adjoint codes for propagating error covariances
in time. However, the trouble of building the codes can be circumvented by using
vi
ensemble filtering techniques, where an ensemble of states is propagated in time using the full nonlinear model, and the statistical information needed in EKF formulas
is estimated from the ensemble. In this paper, we show how the VKF ideas can be
used in the ensemble filtering context. Following VKF, we obtain the state estimate
and its covariance by solving a minimization problem using the limited memory
Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method, which provides low-storage
approximations to the state covariances. The resulting hybrid method, the variational ensemble Kalman filter (VEnKF), has several attractive features compared to
existing ensemble methods. The model error and observation error covariances can
be inserted directly into the minimization problem instead of randomly perturbing
model states and observations as in the standard ensemble Kalman filter. New ensembles can be directly generated from the LBFGS covariance approximation without the need of a square root (Cholesky) matrix decomposition. The frequent resampling from the full state space circumvents the problem of ensemble in-breeding
frequently associated with ensemble filters. Numerical examples are used to show
that the proposed approach performs better than the standard ensemble Kalman filter, especially when the ensemble size is small.
Key Words.
data assimilation, state estimation, Kalman filtering, ensemble filtering, LBFGS
AMS Subject Classifications.
65K10, 15A29
286
Conformal mapping of circular multiply connected domains onto slit domains.
Roman Czapla, Vladimir Mityushev, and Natalia Rylko.
Abstract.
The method of Riemann–Hilbert problems is used to unify and to simplify construction of conformal mappings of multiply connected domains. Conformal mappings
of arbitrary circular multiply connected domains onto the complex plane with slits
of prescribed inclinations are constructed. The mappings are derived in terms of
uniformly convergent Poincaré series. In the proposed method, no restriction on the
location of the boundary circles is assumed. Convergence and implementation of
the numerical method are discussed.
Key Words.
Riemann–Hilbert problem, multiply connected domain, complex plane with slits
AMS Subject Classifications.
30C30, 30E25
298
Optimizing Runge-Kutta smoothers for unsteady flow problems. Philipp Birken.
Abstract.
We consider the solution of unsteady viscous flow problems by multigrid methods employing Runge-Kutta smoothers. Optimal coefficients for the smoothers
are found by considering the unsteady linear advection equation and performing
a Fourier analysis, respectively, looking at the spectral radius of the multigrid iteration matrix. The new schemes are compared to using a classic dual time stepping
approach, where the scheme for the steady state equation is reused, meaning that
vii
the smoother coefficients are obtained by an optimization based on the steady state
equations, showing significant improvements.
Key Words.
multigrid, unsteady flows, finite volume methods, explicit Runge-Kutta methods,
linear advection equation
AMS Subject Classifications.
35L04, 35Q35, 65F10, 65L06, 65M08
313
An iterative substructuring algorithm for a C 0 interior penalty method. Susanne C.
Brenner and Kening Wang.
Abstract.
We study an iterative substructuring algorithm for a C 0 interior penalty method for
the biharmonic problem. This algorithm is based on a Bramble-Pasciak-Schatz preconditioner. The condition number of the preconditioned Schur complement oper2
ator is shown to be bounded by C 1 + ln( H
h ) , where h is the mesh size of the
triangulation, H represents the typical diameter of the nonoverlapping subdomains,
and the positive constant C is independent of h, H, and the number of subdomains.
Corroborating numerical results are also presented.
Key Words.
biharmonic problem, iterative substructuring, domain decomposition, C 0 interior
penalty methods, discontinuous Galerkin methods
AMS Subject Classifications.
65N55, 65N30
333
Cascadic multigrid preconditioner for elliptic equations with jump coefficients.
Zhiyong Liu and Yinnian He.
Abstract.
This paper provides a proof of robustness of the cascadic multigrid preconditioner
for the linear finite element approximation of second order elliptic problems with
strongly discontinuous coefficients. As a result, we prove that the convergence rate
of the conjugate gradient method with cascadic multigrid preconditioner is uniform
with respect to large jumps and mesh sizes.
Key Words.
jump coefficients, conjugate gradient, condition number, cascadic multigrid
AMS Subject Classifications.
65N30, 65N55, 65F10
340
Fejér orthogonal polynomials and rational modification of a measure on the unit
circle. Juan-Carlos Santos-León.
Abstract.
Relations between the monic orthogonal polynomials associated with a measure on
the unit circle and the monic orthogonal polynomials associated with a rational modification of this measure are known. In this paper we deal with some generalization
in order to give an explicit expression of the Fejér orthogonal polynomials on the unit
viii
circle. Furthermore we give a simple and efficient algorithm to compute the monic
orthogonal polynomials associated with a rational modification of a measure.
Key Words.
Fejér kernel, orthogonal polynomials, rational modification of a measure
AMS Subject Classifications.
42C05
353
Locally supported eigenvectors of matrices associated with connected and unweighted power-law graphs. Van Emden Henson and Geoffrey Sanders.
Abstract.
We identify a class of graph substructures that yields locally supported eigenvectors
of matrices associated with unweighted and undirected graphs, such as the various
types of graph Laplacians and adjacency matrices. We discuss how the detection
of these substructures gives rise to an efficient calculation of the locally supported
eigenvectors and how to exploit the sparsity of such eigenvectors to coarsen the
graph into a (possibly) much smaller graph for calculations involving multiple eigenvectors. This preprocessing step introduces no spectral error and, for some graphs,
may amount to considerable computational savings when computing any desired
eigenpair. As an example, we discuss how these vectors are useful for estimating
the commute time between any two vertices and bounding the error associated with
approximations for some pairs of vertices.
Key Words.
graph Laplacian, adjacency matrix, eigenvectors, eigenvalues, sparse matrices
AMS Subject Classifications.
05C50, 05C82, 65F15, 65F50, 94C15
379
A survey and comparison of contemporary algorithms for computing the matrix
geometric mean. Ben Jeuris, Raf Vandebril, and Bart Vandereycken.
Abstract.
In this paper we present a survey of various algorithms for computing matrix geometric means and derive new second-order optimization algorithms to compute the
Karcher mean. These new algorithms are constructed using the standard definition
of the Riemannian Hessian. The survey includes the ALM list of desired properties
for a geometric mean, the analytical expression for the mean of two matrices, algorithms based on the centroid computation in Euclidean (flat) space, and Riemannian
optimization techniques to compute the Karcher mean (preceded by a short introduction into differential geometry). A change of metric is considered in the optimization
techniques to reduce the complexity of the structures used in these algorithms. Numerical experiments are presented to compare the existing and the newly developed
algorithms. We conclude that currently first-order algorithms are best suited for this
optimization problem as the size and/or number of the matrices increase.
Key Words.
matrix geometric mean, positive definite matrices, Karcher mean, Riemannian optimization
AMS Subject Classifications.
15A24, 53B21, 65K10
ix
403
Integrating Oscillatory Functions in M ATLAB, II. L. F. Shampine.
Abstract.
In a previous study we developed a M ATLAB program for the approximation of
Rb
iωx
dx when ω is large. Here we study the more difficult task of approxia f (x)Re
b
mating a f (x) eig(x) dx when g(x) is large on [a, b]. We propose a fundamentally
different approach to the task— backward error analysis. Other approaches require
users to supply the location and nature of critical points of g(x) and may require
g ′ (x). With this new approach, the program quadgF merely asks a user to define the problem, i.e., to supply f (x), g(x), [a, b], and specify the desired accuracy.
Though intended only for modest relative accuracy, quadgF is very easy to use
and solves effectively a large class of problems. Of some independent interest is a
vectorized M ATLAB function for evaluating Fresnel sine and cosine integrals.
Key Words.
quadrature, oscillatory integrand, regular oscillation, irregular oscillation, backward
error analysis, Filon, Fresnel integrals, M ATLAB
AMS Subject Classifications.
65D30, 65D32, 65D07
414
Computation of the matrix pth root and its Fréchet derivative by integrals.
João R. Cardoso.
Abstract.
We present new integral representations for the matrix pth root and its Fréchet
derivative and then investigate the computation of these functions by numerical
quadrature. Three different quadrature rules are considered: composite trapezoidal,
Gauss-Legendre and adaptive Simpson. The problem of computing the matrix pth
root times a vector without the explicit evaluation of the pth root is also analyzed and
bounds for the norm of the matrix pth root and its Fréchet derivative are derived.
Key Words.
matrix pth root, Fréchet derivative, quadrature, composite trapezoidal rule, GaussLegendre rule, adaptive Simpson rule
AMS Subject Classifications.
65F60, 65D30
437
Gradient descent for Tikhonov functionals with sparsity constraints: Theory and
numerical comparison of step size rules. Dirk A. Lorenz, Peter Maass, and
Pham Q. Muoi.
Abstract.
In this paper, we analyze gradient methods for minimization problems arising in the
regularization of nonlinear inverse problems with sparsity constraints. In particular, we study a gradient method based on the subsequent minimization of quadratic
approximations in Hilbert spaces, which is motivated by a recently proposed equivalent method in a finite-dimensional setting. We prove convergence of this method
employing assumptions on the operator which are different compared to other approaches. We also discuss accelerated gradient methods with step size control and
x
present a numerical comparison of different step size selection criteria for a parameter identification problem for an elliptic partial differential equation.
Key Words.
nonlinear inverse problems, sparsity constraints, gradient descent, iterated soft
shrinkage, accelerated gradient method
AMS Subject Classifications.
65K10, 46N10, 65M32, 90C48
464
A multiparameter model for link analysis of citation graphs. Enrico Bozzo and
Dario Fasino.
Abstract.
We propose a family of Markov chain-based models for the link analysis of scientific
publications. The PageRank-style model and the dummy paper model discussed in
[Electron. Trans. Numer. Anal., 33 (2008), pp. 1–16] can be obtained by a particular
choice of its parameters. Since scientific publications can be ordered by the date of
publication it is natural to assume a triangular structure for the adjacency matrix of
the citation graph. This greatly simplifies the updating of the ranking vector if new
papers are added to the database. In addition by assuming that the citation graph
can be modeled as a fixed degree sequence random graph we can obtain an explicit
estimation of the behavior of the entries of the ranking vector.
Key Words.
link analysis, citation graph, random graph, Markov chain, ranking
AMS Subject Classifications.
15B51, 60J20, 65C40
476
On the minimization of a Tikhonov functional with a non-convex sparsity constraint.
Ronny Ramlau and Clemens A. Zarzer.
Abstract.
In this paper we present a numerical algorithm for the optimization of a Tikhonov
functional with an ℓp -sparsity constraints and p < 1. Recently, it was proven that
the minimization of this functional provides a regularization method. We show that
the idea used to obtain these theoretical results can also be utilized in a numerical approach. In particular, we exploit the technique of transforming the Tikhonov
functional to a more viable one. In this regard, we consider a surrogate functional
approach and show that this technique can be applied straightforwardly. It is proven
that at least a critical point of the transformed functional is obtained, which directly
translates to the original functional. For a special case, it is shown that a gradient
based algorithm can be used to reconstruct the global minimizer of the transformed
and the original functional, respectively. Moreover, we apply the developed method
to a deconvolution problem and a parameter identification problem in the field of
physical chemistry, and we provide numerical evidence for the theoretical results
and the desired sparsity promoting features of this method.
Key Words.
sparsity, surrogate functional, inverse problem, regularization
AMS Subject Classifications.
65L09, 65J22, 65J20, 47J06, 94A12
xi